The results of the two preceding chapters, together with the fundamental theorem of Galois theory, suggest that, provided that we can construct enough roots of unity, a separable polynomial is solvable by radicals if and only if its Galois group can be built up in some way from abelian groups. We shall see that this is indeed so.

The rules governing resort to force form a central element within international law, and together with other principles such as territorial sovereignty and the independence and equality of states provide the framework for international order. While domestic systems have, on the whole, managed to prescribe a virtual monopoly on the use of force for the governmental institutions, reinforcing the hierarchical structure of authority and control, international law is in a different situation. It must seek to minimise and regulate the resort to force by states, without itself being able to enforce its will. Reliance has to be placed on consent, consensus, reciprocity and good faith. The role and manifestation of force in the world community is, of course, dependent upon political and other non-legal factors, as well as upon the current state of the law, but the law must seek to provide mechanisms to restrain and punish the resort to violence.

In this chapter we discuss how multiple radios interact. We introduce the concept of duplex and various approaches to enable radios to perform bidirectional communications. We also introduce the concept of network topologies such as star and mesh approaches. We discuss multiple media access control techniques. We introduce aloha, carrier-sense multiple access, time-division multiple access, frequency-division multiple access, and code-division multiple access.

Yes, mathematics may be difficult on occasion, but doing anything technically interesting without math is impossible. Learning mathematics is akin to learning a language, so you should expect that it will take significant practice to become accomplished.

Kenneth Culp Davis, a leading American academic visiting England in the 1960s, described English judicial review as restricted by an ‘old-fashioned, positivist corset astonishing to one with a background in the American legal system’.

State responsibility is a fundamental principle of international law, arising out of the nature of the international legal system and the doctrines of state sovereignty and equality of states. It provides that whenever one state commits an internationally unlawful act against another state, international responsibility is established between the two. A breach of an international obligation gives rise to a requirement for reparation.

What about quintic polynomials? There are quintic polynomials in ℤ[𝑥] which cannot be solved by adjoining radicals. This was proved by Abel in 1824, some five years before the fundamental work of Galois.